So, I figured I'd post another example of something I've written. I think it's one of the better pieces of writing I did at BYU but it's on a topic that most of you will not find at all interesting - the nature of paradox and the applicability of two of the most basic laws of logic. Hope you enjoy!
Excluding Excluded Middle: How to Live in a World of Paradox
Introduction In this paper I will explore two basic laws of propositional logic: the law of excluded middle and the law of non-contradiction. These laws, though fundamental to classical logic, have recently faced challenges. Some logicians challenge the viability of these laws while others insist that they are essential to our understanding. While there are several theories that oppose excluded middle I argue for a view called dialetheism. Dialetheism holds that true contradictions exist. It does not say that all contradictions are true, only that some are (Priest). Among possible true contradictions are logical paradoxes, which have troubled philosophers for centuries. I feel that philosophers have failed to properly resolve the conflicts that arise from these paradoxes. The only way to properly account for these paradoxes is to allow for contradiction. Thus, I will argue that excluded middle and non-contradiction cannot hold as absolute laws.
DilemmaWhat we have is a dilemma: we must either accept non-contradiction and excluded middle or deny them. However, this reasoning requires the use of excluded middle and so difficulties arise. Any time we make a statement of the form “either p or q” we require excluded middle. The word “either” is tricky; it seems to restrict the meaning of the “or” operator. For example, if I were to say “either I will live in Provo or I will live in Pleasant Grove” it seems to restrict the proposition to just those two choices. I can say “I will live in Provo or I will live in Pleasant Grove or I will live in Denver” without a problem, but when I add “either” a third option seems out of place. This restriction of use can lead to fallacy. One of the more common fallacies of deductive reasoning is that of false dilemma. The fallacy of false dilemma is to say something like “you’re either with us or against us” giving only two possible options. It is not true that we are either in agreement “with” a group or “against” that same group; we can be neutral. The United States was effectively neutral during the early years of World War II. If a Frenchman had said “you’re either with us or against us” in reference to the United States, he would have erred; the United States was not fighting along with the French but it was most certainly not fighting against them.
It is easy to find a false dilemma, but the existence of false dilemmas seems to imply the existence of true dilemmas. If true dilemmas exist then the best example of a true dilemma is the law of excluded middle. Excluded middle doesn’t say “you’re either with us or against us” but rather, “you’re either with us or not with us” and works with our example. It could be said that we were not with the French in terms of fighting alongside them and so the example holds. Understanding what constitutes a true dilemma will help us with our dilemma. We should not phrase the problem as “either accept or deny excluded middle and non-contradiction” but rather “either accept or not accept excluded middle and non-contradiction.” If we accept this as a true dilemma we must choose a side of the disjunction.
The LawsExcluded middle and non-contradiction have been recognized since the days of Aristotle. In Book IV of Metaphysics Aristotle says “it is impossible that contradictories should be at the same time true of the same thing,” an early formulation of the law of non-contradiction. He also says that “of one subject we must either affirm or deny any one predicate,” an early formulation of the law of excluded middle. For Aristotle these laws were absolutely vital; he even described non-contradiction as “the most indisputable of all beliefs.” Following Aristotle’s example, most logicians continue to hold these laws as indisputable. In fact, these laws are essential to most forms of classical logic.
Classical logic has not been able to fully represent all of the concepts we want it to. For instance, classical logic can only say that a proposition is either true or false, it cannot account for indeterminacy. But certain propositions are indeterminate, for example, future contingent statements. A future contingent statement is a proposition whose truth value cannot be assessed until the future, for example “I will buy a car next year” might be true or it might be false, we won’t know until next year. Jan Lukasiewiecz attempted represent future contingents by developing a multi-valued logic that provided a truth value of ½ for indeterminate propositions (Bourne 122). This system of logic functions very well with unrelated disjunctions, such as “I will eat breakfast tomorrow or I will jump off a cliff tomorrow.” Unfortunately, it fails to work properly if the disjunctions are contraries, as in “I will eat breakfast tomorrow or I will not eat breakfast tomorrow.” In this case, Lukasiewiecz’s system says that the truth value of the disjunction is indeterminate and, as such, denies excluded middle.
In Future contingents, non-contradiction, and the excluded middle muddle, Craig Bourne argues that there is a way to have this bivalent logic without abandoning the law of excluded middle (125). According to Bourne, the key to maintaining excluded middle in a bivalent system is properly understanding the truth-values of different statements. Bourne’s logic uses 0 to represent falsity, 1 to represent truth, and ½ to represent something that is of indeterminate truth-value. This is similar to Lukasawiecz’s system of representing future conditionals except that Bourne maintains the law of excluded middle. He does this by interpreting negation differently, so that in his truth table ~1/2 (i.e. not indeterminate) is a true statement. Because the proposition says something true the proposition’s has a value of 1 (i.e. true) and not of ½ (i.e. indeterminate). Bourne’s logic seems to work and is representative of a three-valued logic that maintains excluded middle. He accurately represents future contingent statements in a straightforward way. Unfortunately, representing future contingent statements is not the only problem in logic. Bourne recognizes that “this system is not functionally complete” (127). His assertion that “the sorts of truth-functions that cannot be generated by the connectives of this system have no application anyways, and so can be ignored” shows that he neglects possible functions of logic in order to accomplish his goals.
While excluded middle and non-contradiction have been problematic with traditional ideas about logic there are some who have developed logical systems that deny their principles completely. For example, the intuitionist movement in logic is based on premises that do not allow excluded middle to hold. Intuitionists deny the objective existence of mathematical objects, saying that any assertion must be shown by construction (Shapiro 19). This system led to a denial of excluded middle when mathematics evolved to include infinite objects. Posy recognizes this as he says “other logical laws maintained their validity; excluded middle did not” (334). Brouwer, one of the better known intuitionists, claims that classical logic can work for finite domains but fails to account for infinite domains.
For the intuitionist, many of the mistakes of classical mathematics evolved from misunderstandings caused by excluded middle. “Says Brouwer, Hilbert’s oft-repeated slogan that every mathematical problem is ultimately solvable is equivalent to the principle of excluded middle and, as such, amount to an outmoded, ungrounded belief” (Posy 335). According to Brouwer and his intuitionism, classical logic and mathematics rely upon an unfounded belief in excluded middle. Intuitionism recognizes problems in classical logic and mathematics but it chooses to completely reject them, leading to logic and mathematics that are not fully functional.
Excluded middle also faces challenges in classical logic. Perhaps the biggest challenge comes from the inference rule of addition. Addition, or disjunction introduction, is an important inference rule used in predicate logic. While not a law, like excluded middle, addition seems to be a sound rule of inference. But addition and excluded middle cannot both hold. Formalized, excluded middle is “p v ~p”, saying that either p is true or it is not true; it does not allow for any other options but the inference rule of addition does, using addition (and commutation) I can say “p v ~p v q.” As such, there may be a viable third option. We must either change the meaning of our operator “v” or say that excluded middle is not absolute.
Many of the problems I’ve named apply directly to the law of excluded middle, but not to non-contradiction. Formally speaking the two laws are distinct: excluded middle is “p v ~p” and non-contradiction is “~(p ^ ~p).” But according to laws of inference, they are logically equivalent. If I apply De Morgan’s theorem to non-contradiction I get “~p v ~~p” and then through the rules of double-negation and commutation I get excluded middle “p v ~p.” Because the two laws are logically equivalent they are both subject to the same problems. It may be objected that we can resolve the problems I’ve already explained and should not reject fundamental laws without significant evidence. We find significant evidence against these laws when we examine paradoxes.
World of Paradox Paradoxes may be the greatest challenge to these two basic laws of logic. A paradox is “an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises” (Sainsbury 1). The mere notion of paradox appears paradoxical; paradoxes seem to dispute the indisputable. Paradoxes abound, and every true paradox challenges the viability of excluded middle and non-contradiction. I will explore three especially challenging paradoxes.
The first paradox I will explore is the liar. The liar paradox has confounded philosophers for centuries; it is even found in the Bible. Perhaps the most troubling aspect of the liar is its simplicity. Simply put, the liar says “this sentence is false.” If we say that the proposition is true then what it says is false; if we say the proposition is false then what it says is true. This simple proposition challenges the fundamental laws of logic. Because people view excluded middle and non-contradiction as indisputable truths, they struggle to resolve the liar paradox. Some try to resolve it by saying that the statement is neither true nor false, that it has no truth-value (Sorensen 95). On the other hand, some think that the problem with the liar paradox lies in its self-reference (Erickson 109). Unfortunately, every explanation that denies the possibility of contradiction requires us to introduce systems of reasoning that seem counter-intuitive.
Graham Priest argues that paradoxes of this form demonstrate that dialetheias exist (9). The liar and other self-referential paradoxes provide valid and contradictory conclusions from true premises. Priest asserts that if philosophers want to deny dialetheias then they must demonstrate a flaw in this apparently valid reasoning. This usually involves challenging one of the premises of the argument. Not wanting to deny excluded middle or non-contradiction, philosophers respond to the liar and similar paradoxes by explaining complicated issues of self-reference and truth. Some say that the liar is senseless and has no truth-value, while others say that the problem goes away if we just eliminate self-reference (Yablo 251). Unfortunately we can formulate many paradoxes without any need for self-reference (Yablo). And if we claim that some propositions are senseless we must have criteria for doing so beyond saying that they are otherwise paradoxical. In any case, if we follow Occam’s razor the simplest solution is to say that dialetheias exist. This principle holds true with other kinds of paradox as well.
The second paradox I will explore is the heap paradox. The heap paradox is a type of paradox we call a sorites paradox. A sorites is a chain-like argument that uses the conclusion of a set of premises as the premise for the next part of the argument. A sorites paradox uses this method of argumentation to lead to an absurd conclusion. For example, we may have a premise that claims 10,000 grains make a heap of grain (Sainsbury 29). Nobody should reject to such a large pile of grain being called a heap, but if 10,000 grains are a heap then so are 9,999 – removing just one grain will not stop the pile of grain from being a heap. If we continue along this line of reasoning we get: if 9,999 grains make a heap then so do 9,998 grains, and so on. Eventually this line of reasoning will lead to the conclusion that one grain makes a heap. This is obviously wrong but seems to follow from valid reasoning.
While this paradox doesn’t challenge excluded middle and non-contradiction as directly as the liar, it does reveal important information about language. According to excluded middle our collection of grain is either a heap or it is not a heap, but the heap paradox shows us that this is not true. This is because of the vagueness of concepts, like heap. If we reverse the argument we can make a similar chain: one grain is not a heap so two grains are not a heap, and so on until we get to 10,000. This version says that 10,000 grains are not a heap, which is just as wrong as saying that one grain is. There are many concepts that we can only vaguely define. We can’t define a heap as a collection of exactly x number of grains or greater. The law of excluded middle fails to account for this vagueness.
The last paradox I want to examine also challenges excluded middle and non-contradiction through principles of vagueness. Theseus’ paradox examines a ship that belongs to Theseus and has docked for repairs (Olin 4). The ship is made of 1,000 planks and workers renovate the ship, plank by plank. First, they take one plank and move it from the first dock (dock A) to the next dock (dock B), placing it in the exact same position it had on the ship. Next, they replace the first plank with a new plank of wood and continue in this manner until all 1,000 planks have been replaced. In the end there are two ships; one is at dock A and the other is at dock B. But which ship is Theseus’ ship? The ship in dock A begins as Theseus’ ship, and replacing one plank of the ship obviously does not change that. Using a sorites we can conclude that the ship in dock A is Theseus’. In contrast, the ship in dock B has all of the planks that constituted Theseus’ ship arranged in the same manner. As with the heap we can reverse the argument and say that the second ship is Theseus’ ship. We cannot clearly identify which ship is Theseus’ ship.
These three paradoxes use sound reasoning to challenge our notions of non-contradiction and excluded middle. The liar directly challenges the law of non-contradiction, using self-reference to cause a clear case of contradiction. The heap paradox denies the law of excluded middle as it shows that some concepts are vague and that not all things are either have or do not have a given property. Theseus’ paradox compounds the problem introduced with the heap paradox and shows us that concepts are not merely vague but that without accounting for their vagueness we may even have problems with identity. In a world where paradoxes persist, we must either explain how they can be consistent with excluded middle and non-contradiction or we must accept that these laws are not absolute.
Problems of Vagueness The problems with vagueness that the heap paradox and Theseus’ paradox introduce do not only occur in paradox form. Vagueness occurs in many different concepts, e.g. colors, emotions, and tastes. Excluded middle says that something is either white or not white, but what about shades of white? According to excluded middle, an animal is either a bird or not a bird. But how is it that a penguin is a bird and a bat is not a bird? Or more importantly, if I have a dead bird is it still a bird? The arguments behind the heap and Theseus’ paradox can help illustrate the problem. Just as it is impossible to say when a heap becomes a heap it is impossible to determine when a bird is still a bird. If I have a bird without a wing is it still a bird? If it’s missing its head is it still a bird? What if it’s missing both its head and its wings? The law of excluded middle cannot deal with vagueness.
Conclusion
There are good arguments both for and against the law of excluded middle and non-contradiction. The complete acceptance of these laws seems to ignore important paradoxes, inference rules, and principles of language. On the other hand, if I totally deny the law of excluded middle then I deny the plausibility of classical logic and mathematics. I also deny many of the rules I use for basic reasoning. But if I were to wholeheartedly accept these laws then I would deny many alternative logics that also provide great utility and provide us with applicable and valid results. I would also have to explain how non-contradiction and excluded middle can account for vagueness and paradox. It seems we should only conditionally accept the laws. It seems that it is wrong to affirm the law of excluded middle and non-contradiction as either true or false. Like so many other things in this world, these laws are both true and false. As absolute laws they fail to account for important considerations but that does not mean they do not apply at certain times and in certain situations.
The arguments both for and against these fundamental laws convince me. I feel as though the best thing to do is to say that these fundamental laws of logic hold most of the time. In classical logic the dilemma we’ve examined would be formalized as “(p v ~p) v ~(p v ~p).” According to this formalization, even those who deny the law of excluded middle must follow it. You can only conclude that the law of excluded middle does not hold by using the law of excluded middle. I say that there is no middle to exclude here; we do not have a disjunction but a conjunction. It is not “law of excluded middle or not law of excluded middle”, it is “law of excluded middle and not law of excluded middle.” There is value in classical logics and it is impossible to completely deny the law of excluded middle, but it is not absolute. The same is true of non-contradiction, it is wrong to completely deny it. Completely denying non-contradiction would allow absurdities, but completely accepting it allows its own set of absurdities. We must find a way to accept both positions.
The problem with non-contradiction and excluded middle is that they deny a middle ground. According to excluded middle, something either has a property at a given time or it does not. According to non-contradiction, something cannot at the same time be true and false. Unfortunately, paradoxes and natural language contradict non-contradiction and excluded middle. The simplest way to overcome this problem is to deny the absolute nature of these laws. It’s okay that paradoxes contradict the laws, even though it violates those laws. With dialetheism, Graham Priest has proposed a functional system that others have failed to refute. What Aristotle held as indisputable is disputable. The consequences of this position are profound and widespread. In order to accept dialetheism we must shift our paradigm and view the world differently. Completely shifting paradigms is a huge change, but this change can account for paradox and at the same time provide a formal system of reason.
Works Cited
Aristotle. De Interpretatione. 350 B.C.E. Trans. W.D. Ross. The Complete Works of Aristotle.
Ed. Jonathan Barnes. Princeton: Princeton UP, 1984.
Aristotle. Metaphysics. 350 B.C.E. Trans. W.D. Ross. The Complete Works of Aristotle. Ed.
Jonathan Barnes. Princeton: Princeton UP, 1984.
Bourne, Craig. “Future contingents, non-contradiction, and the law of excluded middle muddle.” Analysis. 64.2 (2004): 122-128.
Erickson, Glenn W., and John A. Fossa. Dictionary of Paradox. Lanham, Maryland: University Press of America, 1998.
Olin, Doris. Paradox. Montreal: McGill-Queen’s UP, 2003.
Posy, Carl. “Intuitionism and Philosophy.” The Oxford Handbook of Philosophy of Mathematics and Logic. Ed. Stewart Shapiro. New York: Oxford UP, 2005. 318-55.
Priest, Graham. In Contradiction. 2nd ed. Oxford: Oxford UP, 2006.
Sainsbury, R.M. Paradoxes. 2nd ed. Cambridge: Cambridge UP, 1995.
Shapiro, Stewart. “Philosophy of Mathematics and Its Logic: Introduction.” The Oxford Handbook of Philosophy of Mathematics and Logic. Ed. Stewart Shapiro. New York: Oxford UP, 2005. 3-27.
Sorensen, Roy. A Brief History of the Paradox: Philosophy and the Labyrinths of the Mind. New York: Oxford UP, 2003.
Yablo, Stephen. “Paradox without Self-reference.” Analysis. 53.4 (1993): 251-252.
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